frac·tal (frāk'təl)
n. A geometric pattern that is repeated at ever smaller scales to produce irregular shapes and surfaces that cannot be represented by classical geometry. Fractals are used especially in computer modeling of irregular patterns and structures in nature.
< One of the most famous fractals >
It is known as the Mandelbrot set (M).
Fractal
- So fractal is literally a shape that has repeated patterns and which has a part identical to the whole and whole identical to the part.
(In this case we mean identical as in 'similar'. So they have different sizes)
It may sound difficult but these following pictures will help the understanding.
As you can observe, cows are sprouting from another cow. There are identical cows hanging out of the center cow.
Since we know the basics of the Fractal we shall go in deeper.
- One interesting thing about fractal is unlike other geometric shapes on a euclidean plane, it cannot be easily explained by euclidean geometry.
Also one cool thing about fractal is that the shape can have a finite area while having an infinite perimeter.
As an example: here is the Koch Snowflake.
Having from the left to the right in chronological orders the pattern continues forever. What I meant above by stating that the shape encloses a finite area with an infinite area can be easily explained by the following step.
First of all, this is how to construct the geometric shape. You may follow along if you wish.
1. You need an equilateral prepared to start.
2. You divide each line segment into three segments of equal length.
3. You now draw an equilateral triangle using the middle segment of each sides.
4. Now you have a shape like this: similar to the ones of the Jewish.
Now the thing left to do is to continue the following steps to all of the triangles. And as time passes the triangle will get more and more traingles sticking out and eventually have an infinite perimeter. However the reason why it only has a finite area can be solved by a simple method. It is to draw a circle simply bigger than the triangle.
No matter how many times the step above is repeated, the area of the Koch Snowflake, cannot be bigger than the circle
Here is one real life example from the nature
. As you can notice there are the basic characteristics that I mentioned that are to have as a fractal.spotted in the natural world
Sierpinski’s Triangle
1. Start with a triangle
2. shrink the height and size to 1/2
3. place each point of the shrunk triangle to each mid point of the triangle
4. repeat
CHAOS GAME
1. Take 3 points in a plane to form a triangle, you need not draw it.
2. Randomly select any point inside the triangle.
3. Move half the distance from that point to any of the 3 vertex points.
4. Plot the current position.
5. Repeat from step 3.
3 Dimensional Sierpinski’s Triangle
1. The initial surface area of the tetrahedron, side-length L. At the next process, the side-length is halved
------------------------The equation is L -> L/2
2. there are 4 such smaller tetrahedra. Therefore, the total surface area after the first process is:
3. Repeat the equation, than it will form a shape like this
n. A geometric pattern that is repeated at ever smaller scales to produce irregular shapes and surfaces that cannot be represented by classical geometry. Fractals are used especially in computer modeling of irregular patterns and structures in nature.
< One of the most famous fractals >
It is known as the Mandelbrot set (M).
Fractal
- So fractal is literally a shape that has repeated patterns and which has a part identical to the whole and whole identical to the part.
(In this case we mean identical as in 'similar'. So they have different sizes)
It may sound difficult but these following pictures will help the understanding.
As you can observe, cows are sprouting from another cow. There are identical cows hanging out of the center cow.
Since we know the basics of the Fractal we shall go in deeper.
- One interesting thing about fractal is unlike other geometric shapes on a euclidean plane, it cannot be easily explained by euclidean geometry.
Also one cool thing about fractal is that the shape can have a finite area while having an infinite perimeter.
As an example:
Having from the left to the right in chronological orders the pattern continues forever. What I meant above by stating that the shape encloses a finite area with an infinite area can be easily explained by the following step.
First of all, this is how to construct the geometric shape. You may follow along if you wish.
1. You need an equilateral prepared to start.
2. You divide each line segment into three segments of equal length.
3. You now draw an equilateral triangle using the middle segment of each sides.
4. Now you have a shape like this:
Now the thing left to do is to continue the following steps to all of the triangles. And as time passes the triangle will get more and more traingles sticking out and eventually have an infinite perimeter. However the reason why it only has a finite area can be solved by a simple method. It is to draw a circle simply bigger than the triangle.
No matter how many times the step above is repeated, the area of the Koch Snowflake, cannot be bigger than the circle
Here is one real life example from the nature
.
Sierpinski’s Triangle
1. Start with a triangle
2. shrink the height and size to 1/2
3. place each point of the shrunk triangle to each mid point of the triangle
4. repeat
CHAOS GAME
1. Take 3 points in a plane to form a triangle, you need not draw it.
2. Randomly select any point inside the triangle.
3. Move half the distance from that point to any of the 3 vertex points.
4. Plot the current position.
5. Repeat from step 3.
3 Dimensional Sierpinski’s Triangle
1. The initial surface area of the tetrahedron, side-length L. At the next process, the side-length is halved
------------------------The equation is L -> L/2
2. there are 4 such smaller tetrahedra. Therefore, the total surface area after the first process is:
3. Repeat the equation, than it will form a shape like this